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This paper presents a variational multiscale residual-based stabilized finite element method for the incompressible Navier-Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multiscale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska-Brezzi (inf-sup) condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressurevelocity elements comprising 4-and 10-node tetrahedral elements and 8-and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-driven cavity flow problem.

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This paper presents a variational multiscale stabilized finite element method for the incompressible Navier-Stokes equations. The formulation is written in an Arbitrary Lagrangian-Eulerian (ALE) frame to model problems with moving boundaries. The structure of the stabilization parameter is derived via the solution of the fine-scale problem that is furnished by the variational multiscale framework. The projection of the fine-scale solution onto the coarse-scale space leads to the new stabilized method. The formulation is integrated with a mesh moving scheme that adapts the computational grid to the evolving fluid boundaries and fluid-solid interfaces. Several test problems are presented to show the accuracy and stability of the new formulation.

SUMMARYThis paper presents residual‐based turbulence models for problems with moving boundaries and interfaces. The method is developed via a hierarchical application of variational multiscale ideas and the models are cast in an arbitrary Lagrangian–Eulerian (ALE) frame to accommodate the deformation of domain boundaries. An overlapping additive decomposition of velocity and pressure fields into coarse and fine scale components leads to coarse and fine scale mixed‐field problems. The problem governing fine scales is subjected to a further decomposition of the fine scale velocity into overlapping components termed as fine scales level I and level II. In turn, in the bottom‐up integration of scales, the model for level II fine scales serves to stabilize the problem governing level I fine scales, and model for level I fields yields the turbulence models. From the computational perspective, the coarse scales are represented in terms of the standard Lagrange shape functions, whereas level I and level II scales are represented via quadratic and fourth order polynomial bubbles, respectively. Because of the bubble functions approach employed in the consistently derived fine scale models, the resulting method is free of any embedded or tunable parameters. The proposed turbulence models share a common feature with the LES models in that the largest scales in the flow are numerically resolved, whereas the subgrid scales are modeled. The method is applied to flow around a plunging airfoil at Re = 40,000, and results are compared with experimental and numerical data published in the literature. Also presented are the results for the plunging airfoil at Re = 60,000 to show the robustness and range of applicability of the method. Copyright © 2013 John Wiley & Sons, Ltd.